From Scrooge's study, you can travel to Paterson, New Jersey (1839); Byblos in Phoenicia (ca. 1,000 BC); a paper castle (where you can meet the steadfast tin soldier); and Tharsis, capital of Mars (ca. 2050 AD). Since your goal is apparently to acquire a Colt Paterson revolver, go to Paterson and meet Samuel Colt. You will intuit that he needs to eat something. You will also find a heavy coat there. Take it.
Take the heavy coat to the marketplace in Byblos and give it to Eshak so he can set off on a journey to the north. He will give you an orchid.
Take the orchid to the paper castle and give it to the steadfast tin soldier. He will give you a needle.
Take the needle to the promenade in Tharsis and give it to the custodian. He will give you an eraser.
Take the eraser to the paper castle and give it to the paper dancer. She will give you her spangle.
Take the spangle to Zadar's shop in Byblos and give it to Zadar. She will give you some Tyrian dye.
Take the Tyrian dye to Hadar's shop in Tharsis and give it to Hadar. She will give you some yogurt.
Take the yogurt to New Jersey and give it to Samuel. He will give you a Colt Paterson revolver.
All that remains is to resolve things in the paper castle. Head into the shadowy corridor and talk to the goblin. Ebenezabeth will suggest that she's going to go out into the courtyard and scream. Go out into the courtyard. Scream.
When the game asks you what virtue Ebenezabeth taught to the goblin, notice that "Heavy coat, Orchid, Needle, Eraser, Spangle, Tyrian dye, Yogurt" spells HONESTY.
Play the other games in the Little Match Girl series. They are good.